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        <section id="main"><article id="post-线性代数/实用大众线性代数/线性方程组与矩阵/线性方程组解的几何意义" class="article article-type-post" itemscope itemprop="blogPost">
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      线性方程组解的几何意义
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        <h2 id="二元情况"><a href="#二元情况" class="headerlink" title="二元情况"></a>二元情况</h2><h3 id="唯一解"><a href="#唯一解" class="headerlink" title="唯一解"></a>唯一解</h3><script type="math/tex; mode=display">
\begin{Bmatrix}
 x_1 -  &   x_2 =  &1 \\\\ 
x_1 + & x_2 =  & 3
\end{Bmatrix}</script><p><img src="../assets/4.svg" alt=""></p>
<h3 id="无解-（直线无交点）"><a href="#无解-（直线无交点）" class="headerlink" title="无解 （直线无交点）"></a>无解 （直线无交点）</h3><script type="math/tex; mode=display">
\begin{Bmatrix}
x_1-  & x_2 = &1  \\\\ 
x_1 - &x_2 =  & 3 
\end{Bmatrix}</script><p><img src="../assets/5.svg" alt=""></p>
<h3 id="无穷解-（两直线重合）"><a href="#无穷解-（两直线重合）" class="headerlink" title="无穷解 （两直线重合）"></a>无穷解 （两直线重合）</h3><script type="math/tex; mode=display">
\begin{Bmatrix}
x_1-  & x_2 = &1  \\\\ 
2x_1 - &2x_2 =  & 2 
\end{Bmatrix}</script><p><img src="../assets/6.svg" alt=""></p>
<h3 id="超定二元方程的近似解"><a href="#超定二元方程的近似解" class="headerlink" title="超定二元方程的近似解"></a>超定二元方程的近似解</h3><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">方程组不相容， 没有精确解</span><br></pre></td></tr></table></figure>
<script type="math/tex; mode=display">
\begin{Bmatrix}
x_1 -  & x_2 = &1  \\\\ 
x_1 +  & x_2 = &3  \\\\ 
x_1 + &2x_2 =  & 5 
\end{Bmatrix}</script><p><img src="../assets/7.svg" alt=""></p>
<h2 id="三元情况"><a href="#三元情况" class="headerlink" title="三元情况"></a>三元情况</h2><h3 id="适定方程组-（唯一解）"><a href="#适定方程组-（唯一解）" class="headerlink" title="适定方程组 （唯一解）"></a>适定方程组 （唯一解）</h3><script type="math/tex; mode=display">
\begin{Bmatrix}
x_1 -  & x_2 = &1  \\\\ 
x_1 +  & x_2 = &3  \\\\ 
x_1 + &2x_2 =  & 5 
\end{Bmatrix}</script><script type="math/tex; mode=display">
\left\{\begin{matrix}
x+y-z = 4 \\
2x -3y+z=3 \\
-5x+2y-2z=1\\
\end{matrix}\right.</script><h3 id="消元步骤-（阶梯状）"><a href="#消元步骤-（阶梯状）" class="headerlink" title="消元步骤 （阶梯状）"></a>消元步骤 （阶梯状）</h3><p>$ \left{\begin{matrix}<br>x&amp;+y&amp;-z = 4 \<br>2x&amp; -3y&amp;+z=3 \<br>-5x&amp;+2y&amp;-2z=1\<br>\end{matrix}\right. $&amp;    —-2,3列消去X—-&gt; $ \left{\begin{matrix}<br>x&amp;+y&amp; -z = 4  \<br>&amp;5y&amp; -3z = 5 \<br>&amp;-7y&amp; +7z = -21\<br>\end{matrix}\right. $—-3列消去Y—-&gt; $  \left{\begin{matrix}<br>x&amp;+y&amp;-z = 4  \<br>&amp;5y&amp; - 3z = 5 \<br>&amp;&amp;z = -5\<br>\end{matrix}\right.  $</p>
<p>三平面交于一点</p>
<p><img src="assets/7.png" alt=""></p>
<h3 id="欠定方程组-（有多个解）"><a href="#欠定方程组-（有多个解）" class="headerlink" title="欠定方程组 （有多个解）"></a>欠定方程组 （有多个解）</h3><p> 三平面交于一条直线或者一个平面</p>
<p><img src="assets/8.png" alt=""></p>
<h3 id="不相容-无解"><a href="#不相容-无解" class="headerlink" title="不相容 无解"></a>不相容 无解</h3><p>三个平面不相交 或者 没有公共点，线，面</p>
<figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">线性代数和初等代数的区别:借助于矩阵,用计算机解决问题</span><br></pre></td></tr></table></figure>
      
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              <ol class="toc"><li class="toc-item toc-level-2"><a class="toc-link" href="#二元情况"><span class="toc-number">1.</span> <span class="toc-text">二元情况</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#唯一解"><span class="toc-number">1.1.</span> <span class="toc-text">唯一解</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#无解-（直线无交点）"><span class="toc-number">1.2.</span> <span class="toc-text">无解 （直线无交点）</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#无穷解-（两直线重合）"><span class="toc-number">1.3.</span> <span class="toc-text">无穷解 （两直线重合）</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#超定二元方程的近似解"><span class="toc-number">1.4.</span> <span class="toc-text">超定二元方程的近似解</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#三元情况"><span class="toc-number">2.</span> <span class="toc-text">三元情况</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#适定方程组-（唯一解）"><span class="toc-number">2.1.</span> <span class="toc-text">适定方程组 （唯一解）</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#消元步骤-（阶梯状）"><span class="toc-number">2.2.</span> <span class="toc-text">消元步骤 （阶梯状）</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#欠定方程组-（有多个解）"><span class="toc-number">2.3.</span> <span class="toc-text">欠定方程组 （有多个解）</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#不相容-无解"><span class="toc-number">2.4.</span> <span class="toc-text">不相容 无解</span></a></li></ol></li></ol>
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